Question

Five integers are picked from 0 to 20 with possible representation such that their mean is 12, median is 18 and they have single mode of 20. Ignoring the permutation, the number of ways to pick these five integers is ____.

• A. 2
• B. 1
• C. 3
• D. 0

Hint:

When solving problems with multiple statistical constraints (mean, median, mode), it's often easiest to start with the median and mode, as they fix specific values or frequencies in the dataset. Use the mean condition last to solve for the remaining unknown values.

Answer 1

The Correct Option is B

Step 1: Understanding the Question:
We need to find a unique set of five integers, let's call them $\{a, b, c, d, e\}$, that satisfy four conditions:
1. The integers are between 0 and 20, inclusive.
2. The mean of the five integers is 12.
3. The median is 18.
4. The single mode is 20.
Step 2: Detailed Explanation:
Let's represent the set of five integers in non-decreasing order: $a \le b \le c \le d \le e$.
Condition 3: Median is 18.
The median is the middle value in an ordered set. For five integers, the middle value is the third one, c.
So, $c = 18$. The set is now $\{a, b, 18, d, e\}$.
Condition 4: Single mode of 20.
The mode is the number that appears most frequently. The single mode is 20, which means 20 must appear more than any other number, and it must appear at least twice.
Since the numbers are ordered and the median is 18, the numbers greater than 18 must be 20. To make 20 the mode, both d and e must be 20.
So, $d = 20$ and $e = 20$. The set is $\{a, b, 18, 20, 20\}$.
For 20 to be the *single* mode, no other number can appear twice. This means $a \neq b$, and neither $a$ nor $b$ can be 18.
Condition 2: Mean is 12.
The sum of the five integers is the mean multiplied by the number of integers.
Sum = $12 \times 5 = 60$.
Using our current set:
\[ a + b + 18 + 20 + 20 = 60 \] \[ a + b + 58 = 60 \] \[ a + b = 2 \] Finding a and b:
We need to find two integers, a and b, such that $a+b=2$. We also know from the ordering that $a \le b$, and from the single mode condition, $a \neq b$. Also, $a$ and $b$ must be less than the median 18. The numbers must be between 0 and 20.
The possible pairs of non-negative integers for $(a, b)$ that sum to 2 are (0, 2) and (1, 1).
Case 1: $(a, b) = (1, 1)$. The set would be $\{1, 1, 18, 20, 20\}$. This set has two modes (1 and 20), which violates the "single mode" condition. So, this case is not valid.
Case 2: $(a, b) = (0, 2)$. The set is $\{0, 2, 18, 20, 20\}$. Let's verify all conditions for this set.
- Integers from 0 to 20: Yes.
- Mean: $(0+2+18+20+20)/5 = 60/5 = 12$. Correct.
- Median: The middle number is 18. Correct.
- Single mode: 20 appears twice, all other numbers appear once. Correct.
Step 3: Final Answer:
Only one set of five integers, $\{0, 2, 18, 20, 20\}$, satisfies all the given conditions. Therefore, there is only 1 way to pick these five integers.